Question

A population of values has a normal distribution with μ=151.5 and σ=97.7. You intend to draw a random sample of size n=226. Find the probability that a single randomly selected value is between 148.9 and 169. P(148.9 < X < 169) = Find the probability that a sample of size n=226 is randomly selected with a mean between 148.9 and 169. P(148.9 < M < 169) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

The probability that the sample mean is between 85 and 92 is approximately 0.5888.

Step-by-step explanation:

To find the probability that the sample mean is between 85 and 92, we can follow these steps:

1. Convert the values to z-scores using the formula: z = (x - μ) / (σ / √n).

- For the lower bound, 85: z1 = (85 - 90) / (15 / √25) = -1.

- For the upper bound, 92: z2 = (92 - 90) / (15 / √25) = 0.6667.

2. Use the standard normal distribution table to find the probability between these z-scores. The z-score of -1 corresponds to a probability of approximately 0.1587, and the z-score of 0.6667 corresponds to a probability of approximately 0.7475.

3. To find the probability between these z-scores, subtract the probability corresponding to the lower z-score from the probability corresponding to the upper z-score:

- Probability = 0.7475 - 0.1587 = 0.5888.

Therefore, the probability that the sample mean is between 85 and 92 is approximately 0.5888.